# idempotent-matrix

• A Matrix Equation of a Symmetric Matrix and the Limit of its Solution Let $A$ be a real symmetric $n\times n$ matrix with $0$ as a simple eigenvalue (that is, the algebraic multiplicity of the eigenvalue $0$ is $1$), and let us fix a vector $\mathbf{v}\in \R^n$. (a) Prove that for sufficiently small positive real $\epsilon$, the equation […]
• Normal Nilpotent Matrix is Zero Matrix A complex square ($n\times n$) matrix $A$ is called normal if $A^* A=A A^*,$ where $A^*$ denotes the conjugate transpose of $A$, that is $A^*=\bar{A}^{\trans}$. A matrix $A$ is said to be nilpotent if there exists a positive integer $k$ such that $A^k$ is the zero […]
• Basis of Span in Vector Space of Polynomials of Degree 2 or Less Let $P_2$ be the vector space of all polynomials of degree $2$ or less with real coefficients. Let $S=\{1+x+2x^2, \quad x+2x^2, \quad -1, \quad x^2\}$ be the set of four vectors in $P_2$. Then find a basis of the subspace $\Span(S)$ among the vectors in $S$. (Linear […]
• A Maximal Ideal in the Ring of Continuous Functions and a Quotient Ring Let $R$ be the ring of all continuous functions on the interval $[0, 2]$. Let $I$ be the subset of $R$ defined by $I:=\{ f(x) \in R \mid f(1)=0\}.$ Then prove that $I$ is an ideal of the ring $R$. Moreover, show that $I$ is maximal and determine […]
• Dimension of the Sum of Two Subspaces Let $U$ and $V$ be finite dimensional subspaces in a vector space over a scalar field $K$. Then prove that $\dim(U+V) \leq \dim(U)+\dim(V).$   Definition (The sum of subspaces). Recall that the sum of subspaces $U$ and $V$ is \[U+V=\{\mathbf{x}+\mathbf{y} \mid […]
• Any Finite Group Has a Composition Series Let $G$ be a finite group. Then show that $G$ has a composition series.   Proof. We prove the statement by induction on the order $|G|=n$ of the finite group. When $n=1$, this is trivial. Suppose that any finite group of order less than $n$ has a composition […]